{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:K3FJYW3WUT6IHFIFRN6P5Y7G6A","short_pith_number":"pith:K3FJYW3W","canonical_record":{"source":{"id":"1510.07173","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-24T18:32:08Z","cross_cats_sorted":[],"title_canon_sha256":"f745ad6e08fb0b038aa61b70dfce9348a728b029712048a49261bb1d138a1887","abstract_canon_sha256":"b609dfa23de1539879b1a26baa8c989ab8190547b22b21f24404460feeb3be64"},"schema_version":"1.0"},"canonical_sha256":"56ca9c5b76a4fc8395058b7cfee3e6f019b802d6ef19c6b4b21746558f02989e","source":{"kind":"arxiv","id":"1510.07173","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07173","created_at":"2026-05-18T01:12:07Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07173v1","created_at":"2026-05-18T01:12:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07173","created_at":"2026-05-18T01:12:07Z"},{"alias_kind":"pith_short_12","alias_value":"K3FJYW3WUT6I","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"K3FJYW3WUT6IHFIF","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"K3FJYW3W","created_at":"2026-05-18T12:29:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:K3FJYW3WUT6IHFIFRN6P5Y7G6A","target":"record","payload":{"canonical_record":{"source":{"id":"1510.07173","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-24T18:32:08Z","cross_cats_sorted":[],"title_canon_sha256":"f745ad6e08fb0b038aa61b70dfce9348a728b029712048a49261bb1d138a1887","abstract_canon_sha256":"b609dfa23de1539879b1a26baa8c989ab8190547b22b21f24404460feeb3be64"},"schema_version":"1.0"},"canonical_sha256":"56ca9c5b76a4fc8395058b7cfee3e6f019b802d6ef19c6b4b21746558f02989e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:07.687589Z","signature_b64":"ndWZj7dq53+ddXc63V3fmE2bGozAuNT3NEgq2W1ll+gyrEp4IohNtzBl8Of7SJPs9tZ1RwOXkzKcr+GBslX9CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56ca9c5b76a4fc8395058b7cfee3e6f019b802d6ef19c6b4b21746558f02989e","last_reissued_at":"2026-05-18T01:12:07.687169Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:07.687169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.07173","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SRYr4Zf9m/ypyojCqHOqVgsOANqylehxMIJwOdCKQWHVWCwywccGLWdn/QAhuC2NEhlnadvGqCgdNaUWp7p7Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T00:40:49.675270Z"},"content_sha256":"50b445b7877957a05b3ab04f4810cd0b07047675e17f2b16d4f7761243b4a39a","schema_version":"1.0","event_id":"sha256:50b445b7877957a05b3ab04f4810cd0b07047675e17f2b16d4f7761243b4a39a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:K3FJYW3WUT6IHFIFRN6P5Y7G6A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tobias Black","submitted_at":"2015-10-24T18:32:08Z","abstract_excerpt":"We study nonnnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel whole space system \\begin{align*} \\left\\{\\begin{array}{c@{\\,}l@{\\quad}l@{\\,}c} u_{t}&=\\Delta u-\\nabla\\!\\cdot(u\\nabla v),\\ &x\\in\\mathbb{R}^n,& t>0,\\\\ 0 &=\\Delta v+u+f(x),\\ &x\\in\\mathbb{R}^n,& t>0,\\\\ u(x,0)&=u_{0}(x),\\ &x\\in\\mathbb{R}^n,&\n  \\end{array}\\right. \\end{align*} with prototypical external signal production \\begin{align*} f(x):=\\begin{cases} f_0\\vert x\\vert^{-\\alpha},&\\text{ if }\\vert x\\vert \\leq R-\\rho,\\\\ 0,&\\text{ if } \\vert x\\vert\\geq R+\\rho,\\\\ \\end{cases} \\end{align*} for $R\\in(0,1)$ and $\\rho\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07173","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HwqxVmbAjG2pJCINF7EpDBAXxrgaouvLHDAkYWtGJjkUucKzHpsBTz7VFwgxPgLl8+5KUW0B2LpOXJbpOXaOCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T00:40:49.675645Z"},"content_sha256":"68aa83d4a43d40809207ad5373c428d87e4cb9268623a1cc2a18e30eda69c33b","schema_version":"1.0","event_id":"sha256:68aa83d4a43d40809207ad5373c428d87e4cb9268623a1cc2a18e30eda69c33b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K3FJYW3WUT6IHFIFRN6P5Y7G6A/bundle.json","state_url":"https://pith.science/pith/K3FJYW3WUT6IHFIFRN6P5Y7G6A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K3FJYW3WUT6IHFIFRN6P5Y7G6A/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-22T00:40:49Z","links":{"resolver":"https://pith.science/pith/K3FJYW3WUT6IHFIFRN6P5Y7G6A","bundle":"https://pith.science/pith/K3FJYW3WUT6IHFIFRN6P5Y7G6A/bundle.json","state":"https://pith.science/pith/K3FJYW3WUT6IHFIFRN6P5Y7G6A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K3FJYW3WUT6IHFIFRN6P5Y7G6A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:K3FJYW3WUT6IHFIFRN6P5Y7G6A","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b609dfa23de1539879b1a26baa8c989ab8190547b22b21f24404460feeb3be64","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-24T18:32:08Z","title_canon_sha256":"f745ad6e08fb0b038aa61b70dfce9348a728b029712048a49261bb1d138a1887"},"schema_version":"1.0","source":{"id":"1510.07173","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07173","created_at":"2026-05-18T01:12:07Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07173v1","created_at":"2026-05-18T01:12:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07173","created_at":"2026-05-18T01:12:07Z"},{"alias_kind":"pith_short_12","alias_value":"K3FJYW3WUT6I","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_16","alias_value":"K3FJYW3WUT6IHFIF","created_at":"2026-05-18T12:29:27Z"},{"alias_kind":"pith_short_8","alias_value":"K3FJYW3W","created_at":"2026-05-18T12:29:27Z"}],"graph_snapshots":[{"event_id":"sha256:68aa83d4a43d40809207ad5373c428d87e4cb9268623a1cc2a18e30eda69c33b","target":"graph","created_at":"2026-05-18T01:12:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study nonnnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel whole space system \\begin{align*} \\left\\{\\begin{array}{c@{\\,}l@{\\quad}l@{\\,}c} u_{t}&=\\Delta u-\\nabla\\!\\cdot(u\\nabla v),\\ &x\\in\\mathbb{R}^n,& t>0,\\\\ 0 &=\\Delta v+u+f(x),\\ &x\\in\\mathbb{R}^n,& t>0,\\\\ u(x,0)&=u_{0}(x),\\ &x\\in\\mathbb{R}^n,&\n  \\end{array}\\right. \\end{align*} with prototypical external signal production \\begin{align*} f(x):=\\begin{cases} f_0\\vert x\\vert^{-\\alpha},&\\text{ if }\\vert x\\vert \\leq R-\\rho,\\\\ 0,&\\text{ if } \\vert x\\vert\\geq R+\\rho,\\\\ \\end{cases} \\end{align*} for $R\\in(0,1)$ and $\\rho\\","authors_text":"Tobias Black","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-24T18:32:08Z","title":"Blow-up of weak solutions to a chemotaxis system under influence of an external chemoattractant"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07173","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:50b445b7877957a05b3ab04f4810cd0b07047675e17f2b16d4f7761243b4a39a","target":"record","created_at":"2026-05-18T01:12:07Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"b609dfa23de1539879b1a26baa8c989ab8190547b22b21f24404460feeb3be64","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-24T18:32:08Z","title_canon_sha256":"f745ad6e08fb0b038aa61b70dfce9348a728b029712048a49261bb1d138a1887"},"schema_version":"1.0","source":{"id":"1510.07173","kind":"arxiv","version":1}},"canonical_sha256":"56ca9c5b76a4fc8395058b7cfee3e6f019b802d6ef19c6b4b21746558f02989e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"56ca9c5b76a4fc8395058b7cfee3e6f019b802d6ef19c6b4b21746558f02989e","first_computed_at":"2026-05-18T01:12:07.687169Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:07.687169Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ndWZj7dq53+ddXc63V3fmE2bGozAuNT3NEgq2W1ll+gyrEp4IohNtzBl8Of7SJPs9tZ1RwOXkzKcr+GBslX9CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:07.687589Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.07173","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:50b445b7877957a05b3ab04f4810cd0b07047675e17f2b16d4f7761243b4a39a","sha256:68aa83d4a43d40809207ad5373c428d87e4cb9268623a1cc2a18e30eda69c33b"],"state_sha256":"502504a3554c2e26c53d99b93f6900ec571fe27606ef8cf6c9609ff5ab361ee9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uJtBjs947LT5MGOO/6EQGvyFURxJFhELiWHK3m+GvurFD5Ksw0N5EP4ePL+GMNUGXO4ZT/RmhHe2Dy9dqYFGAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T00:40:49.677597Z","bundle_sha256":"b990133d60e32d9b476fa97fb94845d7983a8126879e66397f9806e359f57e04"}}